1. **Problem statement:** Find the general solution to the differential equation $$y'' - \frac{1}{x} y' + 4x^2 y = 0$$ given one solution $$y_1 = \cos x^2$$ using reduction of order. 2. **Formula and method:** Reduction of order uses the known solution $$y_1$$ to find a second solution $$y_2 = v(x) y_1$$ where $$v(x)$$ is an unknown function. Substitute $$y = v y_1$$ into the differential equation and solve for $$v$$. 3. **Step 1: Write $$y_2 = v(x) \cos x^2$$. Then compute derivatives:** $$y_2' = v' \cos x^2 + v (-2x \sin x^2)$$ $$y_2'' = v'' \cos x^2 + 2 v' (-2x \sin x^2) + v (-2 \sin x^2 - 4x^2 \cos x^2)$$ 4. **Step 2: Substitute $$y_2, y_2', y_2''$$ into the original equation:** $$y_2'' - \frac{1}{x} y_2' + 4x^2 y_2 = 0$$ 5. **Step 3: Simplify using the fact that $$y_1 = \cos x^2$$ is a solution:** Terms involving $$v$$ cancel out, leaving a second order equation in $$v$$: $$v'' \cos x^2 + 2 v' (-2x \sin x^2) - \frac{1}{x} (v' \cos x^2 + v (-2x \sin x^2)) = 0$$ 6. **Step 4: Simplify and rearrange to find an equation for $$v'$$:** After simplification, the equation reduces to: $$v'' \cos x^2 - \frac{1}{x} v' \cos x^2 - 4x v' \sin x^2 = 0$$ 7. **Step 5: Divide through by $$\cos x^2$$ (nonzero) and let $$w = v'$$:** $$w' - \frac{1}{x} w - 4x w \tan x^2 = 0$$ 8. **Step 6: This is a first order linear ODE in $$w$$:** $$\frac{dw}{dx} = \left( \frac{1}{x} + 4x \tan x^2 \right) w$$ 9. **Step 7: Solve by separation of variables:** $$\frac{dw}{w} = \left( \frac{1}{x} + 4x \tan x^2 \right) dx$$ 10. **Step 8: Integrate both sides:** $$\ln |w| = \ln |x| - 2 \ln |\cos x^2| + C$$ 11. **Step 9: Exponentiate:** $$w = v' = C \frac{x}{\cos^2 x^2}$$ 12. **Step 10: Integrate to find $$v$$:** $$v = C \int \frac{x}{\cos^2 x^2} dx$$ 13. **Step 11: Use substitution $$u = x^2$$, $$du = 2x dx$$:** $$v = C \int \frac{x}{\cos^2 x^2} dx = C \frac{1}{2} \int \sec^2 u du = C \frac{1}{2} \tan u + D = C \frac{1}{2} \tan x^2 + D$$ 14. **Step 12: The second solution is:** $$y_2 = v y_1 = \left( C \frac{1}{2} \tan x^2 + D \right) \cos x^2 = C \frac{1}{2} \sin x^2 + D \cos x^2$$ 15. **Step 13: The general solution is:** $$y = c_1 \cos x^2 + c_2 \sin x^2$$ **Answer:** Option D.